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Graphics – courtesy AGI -- celestrak/events/collision/

Graphics – courtesy AGI -- http://www.celestrak.com/events/collision/. The Problem – On-Orbit Collisions …. The Objective of This Work ….

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Graphics – courtesy AGI -- celestrak/events/collision/

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  1. Graphics – courtesy AGI -- http://www.celestrak.com/events/collision/ The Problem – On-Orbit Collisions …

  2. The Objective of This Work … • OBJECTIVE: To develop a Prioritization Methodology for Orbital Debris Removal based on the collisional rate equations presented by Talent (1992) in the context of a simple Particle-In-a-Box (PIB) debris environment evolution model. The discussion in the following slides will develop as follows … • Review of the structure of the single-particle-equivalent PIB model, • Discussion of the requirements for environment stability against catastrophic collisional cascade, • Case 1: Long term evolution illustrated -- without target area removal, • A look at the current LEO population – objects large and small, • Case 2: Long term evolution illustrated -- with target area removal, • Presentation of a collision rate equation for objects of different sizes, • Prioritizing debris objects for removal – discussed and illustrated.

  3. The PIB Model – Described In 1988, inspired in part by the earlier work of Kessler and Cour-Palais (1978), the author of this presentation set out to develop a debris environment model based on a simple differential equation taking an approach that was thermodynamic in character. Originally presented at an AIAA/NASA/DoD conference (1990), the PIB model was later (1992) published as a peer-reviewed paper. Two key elements of the PIB approach are as follows … (1) Treat the entire LEO environment as one equivalent box (having global average characteristics) in which all objects resident might move about, and (2) Describe all objects in the LEO environment in terms of one equivalent particle whose characteristics are determined by the total number of objects, total cross-sectional area, and total mass on orbit at any given moment.

  4. The PIB Model – N, The State Parameter In developing a mathematical model of an evolving system, one must first choose a relevant parameter as the “state” quantity. In the present development, the number of objects, N, resident in the LEO environment at any given time is selected. The reason for this choice is that if an object can be seen, it can be counted – the number of objects on orbit is a direct observable subject, of course, to an appreciation of possible incompleteness, especially at higher altitudes and smaller sizes.

  5. The PIB Model – The Basic Equation The basic PIB differential equation describing the change in the LEO population with time is presented here as … The form of the equation follows from the assumptions that … (1) Deposition reflects the rate, A, at which users of the LEO environment choose to populate it with new objects, (2) Decay due to atmospheric drag and/or random removal may be represented as a finite probability per unit time, B, of the decay (or removal) of any given LEO object, and (3) The production of fragments during collisions, C, between members of the population may be developed in a fashion similar to that for collisions between particles in a gas.

  6. The PIB Model – A: The Deposition Coefficient In the PIB model, the coefficient A is expressed as … where … L = launches per year, worldwide P1 = average number of pieces per launch D1 = fraction of P1 meeting membership conditions FE = fraction of launches eventually resulting in on-orbit non-collisional fragmentation PE = number of fragments produced per explosion DE = fraction of PE meeting membership conditions REM = number of objects removed per year by deliberate retrieval

  7. The PIB Model – B: The Removal Coefficient In the PIB model, the coefficient B is expressed as … where … Batm = reduction fraction per year due to natural drag ( B<0 and B is inversely proportional to D1 – the average population object diameter) S = reduction fraction per year due to the use of a “debris sweeper” (S<0 and may take on any assigned value – typically -0.005 to -0.05) .

  8. The PIB Model – C: The Collision Coefficient In the PIB model, the coefficient C is expressed as the product of two terms … where … d = the number of fragments produced per collision minus the two destroyed during a collision = the collision frequency between members of a population of identical objects (This term is presented explicitly in the next slide.)

  9. The PIB Model – Collision Frequency Between Identical Objects where … Fn= “incomplete mixing factor” – acknowledges that not all objects have access to all parts of the PIB “box” Vc = orbital speed at the average population altitude D1 = the average population object diameter N1= the number of objects having diameter D1 RT= radius – top of the LEO environment shell from Earth’s center RB= radius – bottom of the LEO environment shell from Earth’s center

  10. The PIB Model – Regarding LEO Environmental Stability The form of the PIB environment evolution equation is that of a simple quadratic equation. During the computational execution of the PIB model all relevant parameters are evaluated at the end of each time step including the coefficients A, B, and C as well as the average particle characteristics. Therefore, it is possible to solve for the roots of the PIB equation by application of the quadratic formula shown here as . . . (Note that this author has interchanged the roles of A and C with respect to the usual presentation of the quadratic equation.)

  11. The PIB Model – Regarding LEO Environmental Stability The quantity under the radical is identified as … These terms may be described qualitatively as … The quantity “q” involves the difference between the A and C source terms and the B sink term. Three types of behavior may be identified … : Conditionally Stable : Instability Threshold : Unconditionally Unstable

  12. The PIB Model – Regarding LEO Environmental Stability Consider the illustrative case below for which N1 = 120,000 and N2 = 490,000 … N2 N1 * N As described in this example of “conditionally stable” behavior, the number of objects in LEO, N, always approaches the equilibrium value N1, as long as it is not suddenly perturbed – by accidental or deliberate action – to a state characterized by a value of N > N2 for which dN/dt > 0 for all N.

  13. The PIB Model – Long Term LEO Evolution Illustrated – No Large Object Removal Using “nominal” parameters, the PIB model predicts … (1) The LEO environment is currently unstable and will not exhibit two real roots until 2400. (2) LEO debris growth may be expected to continue until 2590 because the total population, N, will continue to be greater than N2. (3) Throughout the growth period, the average object size will get smaller due primarily to collisions. (4) As the average particle size diminishes, the efficacy of atmospheric drag increases -- N1 will become smaller and N2 will become larger -- eventually overtaking N in 2590. (5) The population reaches a peak of 1,653,000 in 2590 and then begins to decay. (6) Eventually, a stable environment is achieved -- determined by the asymptotic steady-state value of N2 = N = 795,000.

  14. “Target Area” in LEO – NOV. 2009 • DATA SOURCE: NOV2009 RCS data source =CelesTrak (http://celestrak.com/). • OBJECTS OF INTEREST: Resident and/or transient in the altitude regime to 2000 km. • NUMBER OF OBJECTS IDENTIFIED: 12619 with reported RCS values. • SUM TOTAL RCS OF THIS POPULATION: 16122.244 m^2 • AVERAGE CHARACTERISTICS PER PIECE: <area> = 1.2776 m^2 / <D> = 1.2754 m. • STATISTICS – WHERE IS THE COLLISIONAL CROSS-SECTION FOUND? • N Largest Objects % of LEO Population % of “Target” Area • -------------------------- ------------------------------ ---------------------------- • 39 0.31% 10.00% • 126 1.00% 22.00% • 630 5.00% 58.21% • 1262 10.00% 78.59%

  15. Large Object Cross-Sections – Specific Object Types

  16. The PIB Model – Long Term LEO Evolution Illustrated – With Large Object Removal No removal 24 R/B removed per year The removal of two 20 sq-meter rocket bodies (or equivalent) per month can mean the difference between “catastrophic runaway” or relatively benign growth. The next task is then to develop a prioritization list with the objective of removing those objects that are the greatest threat from a future collisions perspective.

  17. Building on The PIB Concept – Multiple Particle Types The development of a multi-particle-type model from the modeling protocol of the original PIB model was described in Talent (1992) and based on the following mathematical prescriptions. The basic PIB, extended to the case of a single environmental box containing m species, becomes … … where the index k may take on values from 1 to m. Regarding the factor, the earlier equation from the PIB is appropriate if i = j; for dissimilar objects the appropriate form is … All notation is similar to the definitions used for the basic PIB model. Perhaps the only additional explanation necessary is to note that d(ij)k means “the number of k-type objects produced during an i-j collision.”

  18. Developing A Debris Removal Prioritization Methodology • ASSERTION: Utilizing elset data and size data (from multiple sources), one may employ the above expression to assess the magnitude of the overall threat posed by each object to the rest of the “targets” in the environment. • PROCEDURAL STEPS: • Let object “1” be the largest member of the population – the calculation of H1,2 may be performed paying particular attention to the calculation of . • Repeat for each member of the population -- H1,3, H1,4, H1,5, H1,6, H1,7, … H1,13000. • Sum over the population to assess the risk associated with this object . • Repeat for every object in the population. • Each object may then be ranked in order of relative threat to the environment. • Thus, the information generated by this analysis will allow for the selection of debris targets for removal achieving maximum benefit per unit cost.

  19. Example Output

  20. The Top 24 R/BS to Remove

  21. Conclusions • The simple PIB model shows there is a threat of catastrophic runaway growth in the orbital debris environment that would not resolve itself for hundreds of years. • Although individual small debris objects are currently the most significant threat to individual high-value assets, the primary threat to the orbital debris environment in the long term may be argued to be the repository of collective collisional cross-sectional area resident in large derelict objects, such as rocket bodies, since they will be the primary source of small debris in the future. • Removing as few as two large derelicts per month can make a significant difference in the future state of the LEO environment. • The collision rate equation can be used as a basis for prioritizing the removal of large objects. A successful demonstration of that has been performed here, suggesting that the production of a strategic decision analysis tool may indeed be worth while.

  22. Dedication Finally, I wish to dedicate this work to my twin sons – Spc. Aaron G. Talent (U.S. Army) and Sgt. Byron K. Talent (U.S. Army) – recent veterans of Operation Iraqi Freedom. Byron Aaron

  23. References Kessler, D.J., and Cour-Palais, B. G., “Collision Frequency of Artificial Satellites: The Creation of a Debris Belt,” Journal of Geophysical Research, Vol. 83, No. A6, pp. 2637-2646, 1978. Talent, D.L., “Analytical Model for Orbital Debris Environment Management,” AIAA/NASA/DoD Orbital Debris Conference: Technical Issues & Future Directions, Paper # AIAA 90-1363, 1990. Talent, D. L., “Analytical Model for Orbital Debris Environmental Management,” Journal of Spacecraft and Rockets, Vol. 29, No. 4, pp. 508-513, July-August 1992.

  24. BACKUP SLIDES

  25. Building on The PIB Concept – Multiple Particle Types / Multiple Strata To extend the PIB model to a n-multistrata system of m-species , (nxm) population evolution equations, , are written. These equations will also include crossfeed terms to accommodate migration from stratum to stratum and/or multistrata/multisize deposition due to fragmentations and collisions. These interactions are illustrated below.

  26. Building on The PIB Concept – Definition of the Multiple Strata In the current version of the PODEM model, ten LEO environmental strata are defined as follows … S1, 350-410 km S6, 690-780 km S2, 410-470 km S7, 780-925 km S3, 470-535 km S8, 925-1180 km S4, 535-605 km S9, 1180-1550 km S5, 605-690 km S10, 1550-2000 km Five discrete object types are defined in PODEM, characterized here by their drag coefficients, Cd(A/M). These values are, specifically … T1, 0.001394; T2, 0.004915; T3, 0.01843; T4, 0.0360, and; T5, 0.1234 Note: T1 and T2 objects -- representative of intact satellites and rocket bodies. T3 and T4 objects -- representative of fragmentation and collision debris large enough to catalog. T5 objects – representative of the small, untracked component of the LEO population.

  27. The PIB Model – “Nominal Parameters” PARAMETERS OF THE MODEL L: LAUNCHES PER YEAR = 70.0000 P1: PIECES PER LAUNCH = 4.1100 D1: FRAC. OF PCS. SURVIVING 1 YR. = 0.6320 FE: FRAC. OF LAUNCHES PROD. EXPL. = 0.0280 DE: FRAC. OF EXPL. ABOVE 350 KM = 0.8200 PE: PIECES PRODUCED PER EXPLOSION = 125.0000 PC: PIECES PRODUCED PER COLLISION = 200.0000 VMIX: FRAC. AVAIL. FOR COLLISION = 0.5500 REM: OBJECTS RETRIEVED PER YEAR = 0.0000 ZHI: TOP OF LEO (KM) = 8378.1348 ZLO: BOTTOM OF LEO (KM) = 6728.1348 VCAVE: ORBITAL SPEED (KM/SEC) = 7.3220 RETEXP: EXPLOSION AREA INC. FACTOR = 6.0000 RETCOL: COLLISION AREA INC. FACTOR = 16.0000 STEP: FRAC. OF YEAR FOR ITER. STEP = 0.0500 ASEL: FREQ., IN YEARS, OF OUTPUT = 10.0000 ZYRS: TOTAL YRS. TO EVOLVE = 8000.0000

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